L(s) = 1 | − 3-s − 7-s − 2·9-s − 11-s − 5·13-s + 7·17-s + 4·19-s + 21-s + 23-s + 5·27-s − 9·29-s − 6·31-s + 33-s + 4·37-s + 5·39-s + 8·41-s − 2·43-s − 5·47-s + 49-s − 7·51-s − 6·53-s − 4·57-s − 6·59-s + 10·61-s + 2·63-s + 16·67-s − 69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 1.38·13-s + 1.69·17-s + 0.917·19-s + 0.218·21-s + 0.208·23-s + 0.962·27-s − 1.67·29-s − 1.07·31-s + 0.174·33-s + 0.657·37-s + 0.800·39-s + 1.24·41-s − 0.304·43-s − 0.729·47-s + 1/7·49-s − 0.980·51-s − 0.824·53-s − 0.529·57-s − 0.781·59-s + 1.28·61-s + 0.251·63-s + 1.95·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9190607333\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9190607333\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.23854057668348, −13.96192886777935, −12.85982220092763, −12.77496252268624, −12.31215478755683, −11.58231233774004, −11.32417934770398, −10.81314253653455, −10.01808436124323, −9.655580246401449, −9.392034317470189, −8.562403979704704, −7.804898723748345, −7.562923964415981, −7.023929916080309, −6.255080475359307, −5.681331945282005, −5.253525317918033, −4.980423840134097, −3.931138978350353, −3.374256520118961, −2.799704465208434, −2.143503544312241, −1.166611295092621, −0.3624416623481253,
0.3624416623481253, 1.166611295092621, 2.143503544312241, 2.799704465208434, 3.374256520118961, 3.931138978350353, 4.980423840134097, 5.253525317918033, 5.681331945282005, 6.255080475359307, 7.023929916080309, 7.562923964415981, 7.804898723748345, 8.562403979704704, 9.392034317470189, 9.655580246401449, 10.01808436124323, 10.81314253653455, 11.32417934770398, 11.58231233774004, 12.31215478755683, 12.77496252268624, 12.85982220092763, 13.96192886777935, 14.23854057668348